Assume that a tunnel is dug through earth from North pole to south pole and that the earth is a non-rotating, uniform sphere of density `rho`. The gravitational force on a particle of mass m dropped into the tunnel when it reaches a distance r from the centre of earth is

Options

A. `(3/(4pi) mG rho) r`
B. `((4pi)/3 mG rho)r`
C. `((4pi)/3 mG rho) r^(2)`
D. `((4pi)/3 m^(2) G rho)r`

Answer

To solve the problem of finding the gravitational force on a particle of mass \( m \) dropped into a tunnel through the Earth at a distance \( r \) from the center, we can follow these steps:

### Step 1: Understand the Setup
We have a tunnel that goes straight through the Earth from the North Pole to the South Pole. The Earth is treated as a non-rotating, uniform sphere with a constant density \( \rho \).

### Step 2: Identify the Relevant Variables
- Let \( R \) be the radius of the Earth.
- Let \( r \) be the distance from the center of the Earth to the particle of mass \( m \).
- The mass of the Earth \( M \) can be expressed in terms of its density \( \rho \).

### Step 3: Calculate the Mass of the Earth
The mass of the Earth can be calculated using the formula for the volume of a sphere:
\[ M = \rho \cdot V = \rho \cdot \frac{4}{3} \pi R^3 \]

### Step 4: Determine the Gravitational Field Inside the Earth
Inside the Earth, the gravitational field \( g' \) at a distance \( r \) from the center is given by:
\[ g' = \frac{G \cdot M'}{r^2} \]
where \( M' \) is the mass of the Earth enclosed within radius \( r \).

### Step 5: Calculate the Enclosed Mass \( M' \)
The mass \( M' \) of the Earth enclosed within radius \( r \) is:
\[ M' = \rho \cdot \frac{4}{3} \pi r^3 \]

### Step 6: Substitute \( M' \) into the Gravitational Field Equation
Substituting \( M' \) into the equation for \( g' \):
\[ g' = \frac{G \cdot \left(\rho \cdot \frac{4}{3} \pi r^3\right)}{r^2} \]
This simplifies to:
\[ g' = \frac{4}{3} \pi G \rho r \]

### Step 7: Calculate the Gravitational Force on the Mass \( m \)
The gravitational force \( F \) acting on the mass \( m \) at distance \( r \) from the center is given by:
\[ F = m \cdot g' = m \cdot \left(\frac{4}{3} \pi G \rho r\right) \]
Thus, the gravitational force is:
\[ F = \frac{4}{3} \pi G \rho m r \]

### Final Answer
The gravitational force on a particle of mass \( m \) dropped into the tunnel when it reaches a distance \( r \) from the center of the Earth is:
\[ F = \frac{4}{3} \pi G \rho m r \]